Multifractional processes with random exponent

Publ. Mat. 49 (2005), 459–486 MULTIFRACTIONAL PROCESSES WITH RANDOM EXPONENT Antoine Ayache and Murad S. Taqqu Abstract Multifractional Processes with Random Exponent (MPRE) are obtained by replacing the Hurst parameter of Fractional Brownian Motion (FBM) with a stochastic process. This process need not be independent of the white noise generating the FBM. MPREs can be conveniently represented as random wavelet series. We will use this type of representation to study their H¨older regularity and their self-similarity. 1. Introduction Fractional Brownian Motion (FBM) with Hurst parameter H∈(0,1), will be denoted {BH(t)}t∈[0,1]. It is well known that, up to a multiplicative constant which only depends on H,BH(t) can be represented through the Wiener integral (1.1) BH(t) = ZR(t−x)H−1/2 +−(−x)H−1/2 +dW(x), where u+= max(u, 0), which is called the moving-average representation of FBM. The measure Wis Gaussian and independently scattered. Another well-known integral representation of FBM is its harmonizable representation, (1.2) BH(t) = ZR eitξ −1 iξ|ξ|H−1/2dc W(ξ). 2000 Mathematics Subject Classification. Primary: 60G18, 60G17; Secondary: 65T16. Key words. H¨older regularity, fractional Brownian motion, self-similarity, sample path properties. This research was partially supported by the NSF Grant DMS-0102410 at Boston University. 460 A. Ayache, M. S. Taqqu The random measures dW and dc Wsatisfy a Parseval type relation, namely for any function f∈L2(R) one has almost surely, (1.3) ZR f(x)dW(x) = ZR ˆ f(ξ)dc W(ξ), where (1.4) ˆ f(ξ) = ZR e−iξ.xf(x)dx is the Fourier transform of f. The complex-valued random measure dc W can therefore be viewed as the Fourier transform of the real-valued random measure dW. Observe that dc Wis completely determined by the Relation (1.3). FBM was introduced in 1940 by Kolmogorov as a way to generate Gaussian “spirals” in Hilbert space [9] and it was made popular by Mandelbrot and Van Ness [12] in 1968. This process has turned out to be very useful in both theory and applications. It has been used to model phenomena in hydrology, economics, finance, physics and telecommunications. For example, Leland, Taqqu, Willinger and Wilson have provided experimental evidence that computer traffic data exhibit long range dependence [10] and since then FBM has been applied with some success as a model in telecommunications (see for instance [15]). The monograph of Doukhan, Oppenheim and Taqqu [6] offers a systematic treatment of FBM, as well as an overview of different areas of applications. One of the main interests of the FBM in modeling is that its H¨older regularity can be prescribed via its Hurst parameter. Actually, αFBM (t), the pointwise H¨older exponent of the FBM at any point t, satisfies almost surely (1.5) αFBM (t, ω) = H, and βFBM (J), its uniform H¨older exponent over an arbitrary non-degenerate interval J, satisfies almost surely (1.6) βFBM (J, ω) = H. The higher the H¨older exponents, the smoother the process. The exponents αand βare defined as follows. Let {X(t)}t∈Tbe a random field with continuous and nowhere differentiable trajectories, defined over a rectangle Tof Rd(that is a set of the form Qd k=1[γk, δk]). The local H¨older regularity of {X(t)}t∈Tin a neighbourhood of each point tcan be measured through its pointwise H¨older exponent, namely Multifractional Processes 461 the stochastic process {αX(t)}t∈Tdefined for every tand ωas, (1.7) αX(t, ω) = sup α, lim sup h→0 |X(t+h, ω)−X(t, ω)| |h|α= 0. The global H¨older regularity of {X(t)}t∈Tover a non-degenerate rectangle J⊂Tcan be measured through its uniform H¨older exponent over J, namely the random variable βX(J) defined for every ωas (1.8) βX(J, ω) = sup β, sup s,s0∈J |X(s, ω)−X(s0, ω)| |s−s0|β<∞. Observe that one always has (1.9) βX(J, ω)≤inf t∈JαX(t, ω). The pointwise and uniform H¨older exponents of the FBM are constant, since they do not depend on the location (i.e. the point tor the interval J) nor ω. This may be undesirable in some situations. For example, FBM is not well adapted to the modeling of non-homogenous materials. Or consider the field of image synthesis: FBM has been frequently used for generating artificial mountains [4], but one obtains in this way mountains whose irregularity is the same everywhere. This is not realistic, since it does not take into account erosion or other meteorological phenomena which smooth some parts of the mountains more than others. The Multifractional Brownian Motion (MBM) has been introduced, independently in [16] and [5], to overcome such limitations of the FBM. Recall that this Gaussian process can be obtained by substituting to the Hurst parameter of the FBM, a function H(·) with values in an arbitrary compact interval [a, b]⊂(0,1). Because of the Relation (1.3) this substitution can be made in the integrals (1.1) or (1.2). Thus the MBM {BH(t)(t)}t∈[0,1] has the following integral representations: (1.10) BH(t)(t) = ZR(t−s)H(t)−1/2 +−(−s)H(t)−1/2 +dW(s) and (1.11) BH(t)(t) = ZR eitξ −1 iξ|ξ|H(t)−1/2dc W(ξ). As in the case of FBM, these representations are respectively called the moving average and the harmonizable representation of the MBM. It has been shown in [16] and in [5], that when βH([0,1]), the uniform H¨older exponent over the interval [0,1] of the function H(·), satisfies the 462 A. Ayache, M. S. Taqqu condition, (1.12) sup t∈[0,1] H(t)< βH([0,1]), then the H¨older regularity of the MBM can be prescribed via H(·). Namely, the pointwise H¨older exponent of the MBM at any point t, satisfies almost surely, (1.13) αMBM (t, ω) = H(t), and the uniform H¨older exponent of the MBM over any non-degenerate interval J⊂[0,1] verifies almost surely, (1.14) βMBM (J, ω) = inf t∈JH(t). Papanicolaou and Sølna [14] have observed that the deterministic functional parameter H(·) of the MBM can be replaced by a stochastic process {S(t)}t∈[0,1]. They suppose, for example, that {S(t)}t∈[0,1] is a stationary process with smooth paths and decaying correlation function that is independent on the white noise, i.e. the Wiener process {W(x)}x∈R which appears for example in Relation (1.1) (see [14, Subsection 4.1, p. 484]). We will call such extensions of the MBM, Multifractional Processes with Random Exponent (MPRE). To define an MPRE we need the following ingredients: • {BH(t)}(t,H)∈[0,1]×[a,b], a Gaussian field with integral representations (1.1) and (1.2). Contrarily to FBM this field depends both on tand H. It is defined over [0,1] ×[a, b] where [a, b]⊂(0,1) is an arbitrary fixed compact interval. • {S(t)}t∈[0,1], a stochastic process with values in [a, b]. Convention. We suppose throughout this paper that 0 < a < b < 1. Definition 1.1. The Multifractional Process with Random Exponent (MPRE) with parameter {S(t)}t∈[0,1] is the stochastic process {Z(t)}t∈[0,1] defined as follows: any trajectory t7→ Z(t, ω) is the composition of the function f1: [0,1] →[0,1]×[a, b], t7→ (t, S(t, ω)) and f2: [0,1]×[a, b]→ R, (t, H)7→ BH(t, ω). Thus, for any t∈[0,1] and ω, one has (1.15) Z(t, ω) = f2(f1(t)) = BS(t,ω)(t, ω). Observe that we do not necessarily suppose that the stochastic parameter {S(t)}t∈[0,1] in the MPRE is independent of the white noise W in (1.10), nor that it is stationary. When {S(t)}t∈[0,1] is independent on the white noise W, the main results on MBM can be extended readily to the MPRE. The more general case, where it can be dependent on the white noise, is more tricky. Actually, in that case, for any non-zero Multifractional Processes 463 fixed t, the process {(t−s)S(t,ω)−1/2 +−(−s)S(t,ω)−1/2 +}s∈R(which depends on the variable s) is no longer adapted to the natural filtration of {W(s)}s∈Rand the MPRE cannot therefore be represented as a usual Itˆo integral. On the other hand, it is possible to adopt the approach of Definition 1.1 because S(t, ω) does not involve the variable s. By using the series representation (2.3) for BH(t), we will, in fact, be working only with sums. The paper is organized as follows. In Section 2, we introduce a wavelet decomposition of the field {BH(t)}(t,H)∈[0,1]×[a,b]and give some properties of this field that will simplify the study of MPREs. Using this wavelet decomposition, we show in Section 3, that the pointwise and uniform H¨older exponents of the MPRE can be prescribed by its random parameter {S(t)}t∈[0,1]. Finally, in Section 4, we give sufficient conditions for the MPRE to be self-similar (in the sense of marginal distributions) or have stationary increments. A word about notation. We will be using non-random as well as random constants. To ease the distinction, we use small letters (e.g. c) to denote non-random constants and capital letters (e.g. C=C(ω)) to denote random constants. Moreover, for the sake of simplicity, the stochastic processes considered here are real-valued and often defined on the interval [0,1]. Our results remain true when the interval [0,1] is replaced by a compact cube of Rd. 2. Some useful properties of the random field {BH(t)}(t,H)∈[0,1]×[a,b] We obtain, in this section, some properties of the field {BH(t)}(t,H)∈[0,1]×[a,b]that will simplify the study of the MPRE {Z(t)}t∈[0,1]. Recall that {BH(t)}(t,H)∈[0,1]×[a,b]and {Z(t)}t∈[0,1] are related through Relation (1.15). We first provide a wavelet decomposition of the field {BH(t)}(t,H)∈[0,1]×[a,b]. Let {ψj,k(x)}(j,k)∈Z2be a Lemari´e-Meyer wavelet basis of the Hilbert space L2(R). Recall that such basis has the following properties. (P1) The functions ψj,k are generated by dilations and translations of a unique function ψcalled a mother wavelet. Namely, for every j∈Z, k∈Zand x∈R, one has (2.1) ψj,k(x) = 2j/2ψ(2jx−k), 464 A. Ayache, M. S. Taqqu or equivalently b ψj,k, the Fourier transform of ψj,k, satisfies for every ξ∈R, (2.2) b ψj,k(ξ) = 2−j/2e−ik2−jξb ψ(2−jξ). In addition, the functions ψj,k, and consequently the functions b ψj,k, belong to the Schwartz class S(R). Recall that S(R) is the space of all infinitely differentiable functions uwhose derivatives u(n)of any order n≥0 satisfy for all integer m, lim |t|→∞ tmu(n)(t) = 0. Observe that the tails of u(n)decrease faster than any polynomial. (P2) For any integers jand kthe support of b ψj,k is contained in the domain nξ∈R,2j+1π 3≤ |ξ| ≤ 2j+3π 3o. (P3) Up to a multiplicative factor 1/√2πthat we will neglect, {b ψj,k(−ξ)}(j,k)∈Z2forms an orthonormal basis of L2(R). To obtain a random wavelets series representation of the field {BH(t)}(t,H)∈[0,1]×[a,b], one fixes (t, H)∈[0,1]×[a, b], and decomposes its kernel, namely the function b f:ξ7→ eitξ−1 iξ|ξ|H−1/2in the basis {b ψj,k(−ξ)}j,k∈Z. Then one applies the integral RRb f(·)dc Wto this decomposition. Since, this integral is an isometry from the Hilbert space L2(R) into the Hilbert space L2(Ω) of the square integrable and mean-zero random variables, one obtains, in view of (1.2), that for every (t, H)∈[0,1] ×[a, b], (2.3) BH(t) = X j∈ZX k∈Z aj,k(t, H)j,k, where {j,k}j,k∈Zis a sequence of N(0,1) Gaussian random variables and where the non-random coefficients aj,k(t, H) are given by (2.4) aj,k(t, H) = ZR eitξ −1 iξ|ξ|H−1/2b ψj,k(ξ)dξ. We will show later that the series (2.3) is, with probability 1, uniformly convergent in (t, H). If we define, for every (x, H)∈R×[a, b], the function (2.5) Ψ(x, H) = ZR eixξ b ψ(ξ) iξ|ξ|H−1/2dξ, then by setting η= 2−jξin the integral (2.4) and using (2.2), one gets (2.6) aj,k(t, H) = 2−jH(Ψ(2jt−k, H)−Ψ(−k, H)). Multifractional Processes 465 Observe that the integral (2.5) converges since b ψbelongs to S(R) and vanishes in a neighbourhood of the origin. We now give some useful properties of the function Ψ. Lemma 2.1. Ψis a C∞function over R×[a, b]and its partial derivatives of any order are localized in the variable xuniformly in the variable H. As a consequence, for all integers mand nthere is a constant c > 0(that only depends on m,n,aand b) such that for every (x, H)∈R×[a, b] one has, (2.7)  ∂m+n (∂x)m(∂H)nΨ(x, H)≤c(2 + |x|)−2. Proof: First we will suppose that x≥0. Let Kbe the integrand in (2.5), namely the function defined for any (x, H, ξ)∈R+×[a, b]×R, as K(x, H, ξ) = eixξ b ψ(ξ) iξ|ξ|H−1/2.Kis an infinitely differentiable function in (x, H), whose partial derivatives of any order are bounded uniformly in (x, H) by a ξintegrable function. It follows that for every integers m, nand (x, H)∈R+×[a, b], one has ∂m+n (∂x)m(∂H)nΨ(x, H) = ZR ∂m+n (∂x)m(∂H)nK(x, H, ξ)dξ, which implies that  ∂m+n (∂x)m(∂H)nΨ(x, H)=ZR ei(2+x)ξb φ(ξ) iξ|ξ|H−1/2dξ, where b φis the function of S(R) defined for every real ξby b φ(ξ) = e−i2ξξm(log |ξ|)nb ψ(ξ). Then integrating twice by parts, one obtains that ZR ei(2+x)ξb φ(ξ) iξ|ξ|H−1/2dξ≤(2 + x)−2ZR |b φ00(ξ)| |ξ|H+1/2+(2H+ 1) |b φ0(ξ)| |ξ|H+3/2 + (H+ 1/2)(H+ 3/2) |b φ(ξ)| |ξ|H+5/2!dξ ≤c(2 + x)−2, where the constant c=5RR P2 p=0 |b φ(2−p)(ξ)|1 |ξ|a+1/2+p+1 |ξ|b+1/2+pdξ. The case where x < 0 can be treated similarly. 466 A. Ayache, M. S. Taqqu Let {˙ BH(t)}(t,H)∈R×[a,b]and {¨ BH(t)}(t,H)∈R×[a,b]be respectively the low frequency and the high frequency components of the wavelet representation of {BH(t)}(t,H)∈R×[a,b]. These fields are defined for every (t, H)∈R×[a, b] as, ˙ BH(t) = −1 X j=−∞ X k∈Z 2−jH j,k(Ψ(2jt−k, H)−Ψ(−k, H)) = ∞ X j=1 X k∈Z 2jH −j,k(Ψ(2−jt−k, H)−Ψ(−k, H)), (2.8) and (2.9) ¨ BH(t) = ∞ X j=0 X k∈Z 2−jH j,k(Ψ(2jt−k, H)−Ψ(−k, H)). Clearly, one has (2.10) BH(t) = ˙ BH(t) + ¨ BH(t). We now provide some properties of {˙ BH(t)}(t,H)∈R×[a,b]and {¨ BH(t)}(t,H)∈R×[a,b]. The proofs of the following propositions will be given at the end of this section. Proposition 2.1. The trajectories of the field {˙ BH(t)}(t,H)∈R×[a,b]are with probability 1,C∞functions over R×[a, b]. Proposition 2.2. There is an event Ω∗ 1of probability 1, satisfying the following properties: (a) For any ω∈Ω∗ 1, the function (t, H)7→ ¨ BH(t, ω)is continuous over [0,1] ×[a, b]. (b) For all ω∈Ω∗ 1and all reals mand Msuch that a≤m≤M≤b, the uniform H¨older exponent of the function (t, H)7→ ¨ BH(t, ω) over the rectangle [0,1]×[m, M]is equal to m(see Relation (1.8) for the definition of this exponent). This means that for any arbitrarily small  > 0, there is a random variable C1>0(which only depends on m,Mand ) such that the inequality (2.11) |¨ BH0(t0, ω)−¨ BH00 (t00, ω)| ≤ C1(ω)(|t0−t00|+|H0−H00|)m−, holds for all ω∈Ω∗ 1,(t0, H0)∈[0,1] ×[m, M]and (t00, H00)∈ [0,1] ×[m, M]. Multifractional Processes 467 (c) For all ω∈Ω∗ 1, the function H7→ ¨ BH(t, ω)is Lipschitz over [a, b] uniformly in t∈[0,1]. More precisely, there is a random variable C2>0(which only depends on aand b) such that for all H0∈ [a, b]and H00 ∈[a, b]one has (2.12) sup t∈[0,1] |¨ BH0(t, ω)−¨ BH00 (t, ω)| ≤ C2(ω)|H0−H00|. The following theorem is a straightforward consequence of Propositions 2.1 and 2.2. Theorem 2.1. Proposition 2.2 remains true when the high-frequency field {¨ BH(t)}(t,H)∈[0,1]×[a,b]is replaced by the field {BH(t)}(t,H)∈[0,1]×[a,b]. We now state two lemmas that we need in the proofs of Propositions 2.1 and 2.2. The first lemma follows from the Borel-Cantelli lemma and one may refer to [13] or [3] for example for its proof. The second lemma is a reformulation of a strong version of Kolmogorov-Centsˇov criterion (see Chapter 2 of [8]). The proof of that lemma can be found in e.g. [2]. Lemma 2.2. There is a random variable C3>0with finite moments of any order and there is an event Ω∗ 3of probability 1, such that for any ω∈Ω∗ 3,j∈Zand k∈Zone has (2.13) |j,k(ω)| ≤ C3(ω)plog(2 + |j|)plog(2 + |k|). Lemma 2.3. Let {X(τ)}τ∈Tbe a Gaussian field with continuous trajectories defined on a rectangle Tof Rd(i.e. a set of the form Qd k=1[γk, δk]). Suppose that for some constants µ∈(0,1) and c4>0, the inequality (2.14) E(|X(τ0)−X(τ00)|2)≤c4|τ0−τ00|2µ, holds for every τ0, τ00 ∈T, where |·|is an arbitrary norm on Rd. Then, the uniform H¨older exponent over Tof the field {X(τ)}τ∈Tis almost surely greater than µ. This means that there is Ω∗ 4, an event of probability 1, that only depends on T, such that for every ω∈Ω∗ 4and every arbitrarily small real  > 0, the inequality |X(τ0, ω)−X(τ00, ω)| ≤ C4(ω)|τ0−τ00|µ−, holds for all τ0∈Tand τ00 ∈T(observe that the random variable C4 only depends on ). Proof of Proposition 2.1: In view of Lemma 2.1, it is sufficient to show that the series (2.8) and all the series obtained by differentiating it term by term are, with probability 1, uniformly convergent in the variable (t, H) on each compact set of the form [−d, d]×[a, b] where the 474 A. Ayache, M. S. Taqqu Lemma 3.2. Fix t0∈[0,1] and ω∈Ω∗ 3, the event of probability 1 introduced in Lemma 2.2. Then the series (3.11) ¨ Tt0(t, ω) = ∞ X j=0 X k∈Z 2−jS(t0,ω)j,k(ω)Ψ(2jt−k, S(t0, ω)), is uniformly convergent in ton every compact of R. In addition, there is a random variable C5>0such that for every t∈Rand ω∈Ω∗ 3, one has (3.12) |¨ Tt0(t, ω)| ≤ C5(ω)plog(2 + |t|). Proof of Lemma 3.2: To show that the series (3.11) is uniformly convergent in ton every compact of Rone uses the inequalities (2.7) and (2.13) and the same techniques as in Section 3 in [3]. Let us now prove that the inequality (3.12) holds. Using again the inequalities (2.7) and (2.13) one obtains that for every t∈R (3.13) |¨ Tt0(t, ω)| ≤ C(ω) ∞ X j=0 2−jS(t0,ω)log1/2(2+j)X k∈Z log1/2(2 + |k|) (2 + |2jt−k|)2. Let [2jt] denote the integer part of 2jt. Using the sub-additivity of the function x7→ log1/2(2 + x) (see Lemma 3.5 of [3]) and the inequality a+b≤ab which holds for all reals a≥2 and b≥2, we have that X k∈Z log1/2(2 + |k|) (2 + |2jt−k|)2≤X k∈Z log1/2(9 + |k+ [2jt]|) (2 + |2jt−[2jt]−k|)2 ≤X k∈Z log1/2(9 + |k|) (2 + |2jt−[2jt]−k|)2 +X k∈Z log1/2(9 + 2j) (2 + |2jt−[2jt]−k|)2 +X k∈Z log1/2(9 + |t|) (2 + |2jt−[2jt]−k|)2 ≤c0log1/2(9 + 2j) log1/2(9 + |t|), (3.14) where the constant c0= 2 supx∈[0,1] Pk∈Z log1/2(9+|k|) (2+|x−k|)2. Thus (3.12) follows from (3.13) and (3.14). Multifractional Processes 475 Lemma 3.3. Fix t0∈(0,1) and let {α¨ Tt0(t)}t∈Rand {α¨ Z(t)}t∈(0,1) be respectively the pointwise H¨older exponents of the processes {¨ Tt0(t)}t∈R and {¨ Z(t)}t∈(0,1). Then for any ω∈Ω∗ 1, the event with probability 1that has been introduced in Proposition 2.2, one has (3.15) α¨ Z(t0, ω)≤S(t0, ω)if and only if α¨ Tt0(t0, ω)≤S(t0, ω). Proof of Lemma 3.3: Since βS([0,1]), the uniform H¨older exponent over [0,1] of the process {S(t)}t∈[0,1], satisfies the condition (C), it is sufficient to prove that for any ω∈Ω∗ 1and any αin the interval 0, βS([0,1], ω) one has lim sup h→0 |¨ Tt0(t0+h, ω)−¨ Tt0(t0, ω)| |h|α= 0(3.16) if and only if lim sup h→0 |¨ Z(t0+h, ω)−¨ Z(t0, ω)| |h|α= 0.(3.17) It follows from Definition 3.1 that for any t0+h∈[0,1], one has |¨ BS(t0,ω)(t0+h, ω)−¨ BS(t0,ω)(t0, ω)| −|¨ BS(t0+h,ω)(t0+h, ω)−¨ BS(t0,ω)(t0+h, ω)| ≤ |¨ Z(t0+h, ω)−¨ Z(t0, ω)| ≤ |¨ BS(t0,ω)(t0+h, ω)−¨ BS(t0,ω)(t0, ω)| +|¨ BS(t0+h,ω)(t0+h, ω)−¨ BS(t0,ω)(t0+h, ω)|. (3.18) On the one hand, using the Relation (2.12) and the definition of the uniform H¨older exponent we obtain that |¨ BS(t0+h,ω)(t0+h, ω)−¨ BS(t0,ω)(t0+h, ω)| ≤sup t∈[0,1] |¨ BS(t0+h,ω)(t, ω)−¨ BS(t0,ω)(t, ω)| ≤C2(ω)|S(t0+h, ω)−S(t0, ω)| ≤C(ω)|h|βS([0,1],ω)−, (3.19) where the real  > 0 is arbitrarily small. On the other hand, Relations (2.9) and (3.11) imply that (3.20) |¨ Tt0(t0+h, ω)−¨ Tt0(t0, ω)|=|¨ BS(t0,ω)(t0+h, ω)−¨ BS(t0,ω)(t0, ω)|. 476 A. Ayache, M. S. Taqqu Thus, it follows from (3.18), (3.19) and (3.20) that |¨ Tt0(t0+h, ω)−¨ Tt0(t0, ω)| −C(ω)|h|βS([0,1],ω)− ≤ |¨ Z(t0+h, ω)−¨ Z(t0, ω)| ≤ |¨ Tt0(t0+h, ω)−¨ Tt0(t0, ω)| +C(ω)|h|βS([0,1],ω)−, which proves that the Relations (3.16) and (3.17) are equivalent. Lemma 3.4. Let e Ψbe a function defined for every (x, H)∈R×[a, b] as (3.21) e Ψ(x, H) = iZR eixξξ|ξ|H−1/2b ψ(ξ)dξ. This function has the following properties: (i) e Ψis localized in the variable xuniformly in the variable H. More precisely, there is a constant c > 0(that only depends on aand b) such that for any (x, H)∈R×[a, b]one has (3.22) |e Ψ(x, H)| ≤ c(2 + |x|)−2. (ii) For any H∈[a, b], the first moment of the function e Ψ(., H)vanishes, that is, (3.23) ZRe Ψ(x, H)dx = 0. (iii) Let Ψbe the function introduced in (2.5). For any H∈[a, b], the system of functions {2j/2Ψ(2jt−k, H); j∈Nand k∈Z}and {2j/2e Ψ(2jt−k, H); j∈Nand k∈Z}is biorthogonal. This means that for any j∈N,j0∈N,k∈Zand k0∈Z, one has (3.24) 2(j+j0)/2ZR Ψ(2jt−k, H)e Ψ(2j0t−k0, H)dt =δ(j, k;j0, k0), where δ(j, k;j0, k0) = 1 if (j, k) = (j0, k0)and 0otherwise. Proof of Lemma 3.4: Part (i) can be shown as in Lemma 2.1. To prove Parts (ii) and (iii) let us first observe that for every H∈[a, b] the Fourier transforms of the functions x7→ Ψ(x, H) and x7→ e Ψ(x, H) are respectively the functions ξ7→ b ψ(ξ) iξ|ξ|H−1/2and ξ7→ iξ|ξ|H−1/2b ψ(ξ). As ξ7→ iξ|ξ|H−1/2b ψ(ξ) vanishes in a neighbourhood of the origin one Multifractional Processes 477 gets (ii). Part (iii) can be obtained as follows. Using Parseval Formula we have that 2(j+j0)/2ZR Ψ(2jt−k, H)e Ψ(2j0t−k0, H)dt = 2−(j+j0)/2ZR e−i(k/2j−k0/2j0)ξ|2−j0ξ|H+1/2 |2−jξ|H+1/2b ψ(2−jξ)b ψ(2−j0ξ)dξ = 2(j−j0)(H+1/2) ×2−(j+j0)/2ZR e−i(k/2j−k0/2j0)ξb ψ(2−jξ)b ψ(2−j0ξ)dξ = 2(j−j0)(H+1/2) ×2(j+j0)/2ZR ψ(2jt−k)ψ(2j0t−k0)dt =δ(j, k;j0, k0). Observe that the last equality follows from the orthonormality of the functions 2j/2ψ(2jt−k) (see the beginning of Section 2). Now we are able to prove Theorems 3.1 and 3.2. Proof of Theorem 3.1: Since the function t7→ S(t, ω) is continuous over the interval [0,1] with probability 1, there is, for any t0∈(0,1) and any arbitrarily small  > 0, an η > 0, such that inf t∈[t0−η,t0+η]S(t, ω)≥S(t0, ω)−. Lemma 3.1 and inequality (1.9) thus imply that almost surely αZ(t0, ω)≥S(t0, ω)−. Letting →0 we obtain that, with probability 1, αZ(t0, ω)≥S(t0, ω). Let us now show that, almost surely, αZ(t0, ω)≤S(t0, ω). It follows from Remark 3.2 and Lemma 3.3 that one can prove instead (3.25) α¨ Tt0(t0, ω)≤S(t0, ω). To establish this last inequality, suppose ad absurdum that there is a non-negligible event Asuch that for any ω∈Aone has (3.26) α¨ Tt0(t0, ω)> S(t0, ω). We will use a method which allowed Jaffard [7] to obtain a wavelet characterization of the pointwise H¨older exponent. Let Ω∗ 3be the event 478 A. Ayache, M. S. Taqqu of probability 1 that has been introduced in Lemma 2.2. Relations (3.12) and (3.22) imply that for any j∈N,k∈Zand ω∈Ω∗ 3∩A, the integral (3.27) Ij,k(ω) = 2jZR ¨ Tt0(t, ω)e Ψ(2jt−k, S(t0, ω)) dt, is convergent and Relations (3.11) and (3.24) imply that (3.28) Ij,k(ω) = 2−jS(t0,ω)j,k(ω). Choose η > 0 such that (3.29) S(t0, ω) + η < min(α¨ Tt0(t0, ω),1). Using the definition of the pointwise H¨older exponent, namely Relation (1.7) one obtains that for any |t−t0|small enough (3.30) |¨ Tt0(t, ω)−¨ Tt0(t0, ω)| ≤ C(ω)|t−t0|S(t0,ω)+η and thanks to Relation (3.12) this inequality remains true for any t∈R. Then it follows from Relations (3.22), (3.23), (3.27) and (3.30) that for any j∈N,k∈Zand ω∈Ω∗ 3∩A, one has |Ij,k(ω)|= 2jZR (¨ Tt0(t, ω)−¨ Tt0(t0, ω))e Ψ(2jt−k, S(t0, ω)) dt ≤2jZR|¨ Tt0(t, ω)−¨ Tt0(t0, ω)||e Ψ(2jt−k, S(t0, ω))|dt ≤C0(ω)2jZR |t−t0|S(t0,ω)+η (2 + |2jt−k|)2dt. Setting u= 2jt−kin this last integral yields |Ij,k(ω)| ≤ C0(ω)ZR |2−j(u+k)−t0|S(t0,ω)+η (2 + |u|)2du. Since |2−j(u+k)−t0|S(t0,ω)+η≤ |2−ju|S(t0,ω)+η+|t0−2−jk|S(t0,ω)+η, one obtains (3.31) |Ij,k(ω)| ≤ C00(ω)2−j(S(t0,ω)+η)(1 + |2jt0−k|S(t0,ω)+η), where C00(ω) = C0(ω)RR |u|S(t0,ω)+η (2+|u|)2du +RR du (2+|u|)2. Relations (3.28) and (3.31) imply (3.32) |j,k(ω)| ≤ C00(ω)2−jη(1 + |2jt0−k|S(t0,ω)+η), for all ω∈Ω∗ 3∩A,j∈Nand k∈Z. Suppose now that n7→ (jn, kn) is a sequence with values in N×Zsatisfying for all n∈N,jn≥n and |2jnt0−kn| ≤ 1. To simplify our notations we set n=jn,knfor Multifractional Processes 479 every n∈N. Observe that the inequality (3.32) entails that for all n∈N and ω∈Ω∗ 3∩A, (3.33) |n(ω)| ≤ C00(ω)2−nη+1. On the other hand, since for each n,nis a standard random Gaussian variable, one has P(|n| ≥ 1) >0 and thus P∞ n=0 P(|n| ≥ 1) = ∞. Since the random variables nare independent, the Borel-Cantelli lemma implies that there is Ω∗ 6, an event of probability 1, with the following property: for any ω∈Ω∗ 6there is a subsequence l7→ nlsuch that for every l, (3.34) |nl(ω)| ≥ 1. Taken together, Relations (3.33) and (3.34) imply that for any l∈N and ω∈Ω∗ 3∩A∩Ω∗ 6one has 1 ≤C00(ω)2−nlη+1, which is a contradiction. Proof of Theorem 3.2: This theorem follows from Theorem 3.1, Relation (1.9) and Lemma 3.1. Proposition 3.2. If Sis a non-degenerate random process, then the resulting MPRE is a non-Gaussian stochastic process. This follows from Theorem 3.1 and the following lemma. Lemma 3.5. If {X(t)}t∈[0,1] is a Gaussian process with continuous and nowhere differentiable trajectories, then αX(t), its pointwise H¨older exponent at an arbitrary t, is almost surely deterministic. Proof of Lemma 3.5: Let 0 < s1≤s2be such that (3.35) P(s1≤αX(t)) >0 and P(αX(t)≤s2)>0, and for k∈ {1,2}, let {Yt,k(u)}u∈[0,1] be the Gaussian process defined as (3.36) Yt,k(u) =      0,if u=t X(u)−X(t) |u−t|sk,otherwise. It follows from Relation (1.7) and the continuity of the process {X(t)}t∈[0,1] that one has P(supu∈[0,1] |Yt,1(u)|<∞)>0 and P(supu∈[0,1] |Yt,2(u)|=∞)>0. Then using Proposition 1, p. 211 of [11] one gets that P(supu∈[0,1] |Yt,1(u)|<∞) = 1 and P(supu∈[0,1] |Yt,2(u)|= ∞) = 1. This means that P(s1≤αX(t)) = 1 and P(αX(t)≤s2) = 1. Set now (3.37) h(t) = sup{s1>0; P(s1≤αX(t)) = 1}. 480 A. Ayache, M. S. Taqqu Then for every s2> h(t) one has P(αX(t)≤s2) = 1 and hence (3.38) h(t) = inf{s2;P(αX(t)≤s2) = 1}. Relations (3.37) and (3.38) imply that with probability 1, αX(t) = h(t). 4. Self-similarity and stationarity of the increments of some classes of MPRE In this section, we give sufficient conditions for the MPRE to be selfsimilar (in the sense of marginal distributions) or have stationary increments. In the following theorem we suppose for convenience that the MPRE is defined on the whole real line and not only on the interval [0,1]. Theorem 4.1. Let {Z(t)}t∈Rbe an MPRE whose parameter {S(t)}t∈R is a stationary stochastic process independent of the white noise. Then {Z(t)}t∈Rsatisfies the following self-similarity property. For any reals a > 0and t, one has (4.1) Z(at)(d1) =aS(t)Z(t), where (d1) =means equality of the marginal distributions. To prove this theorem we use Auscher’s wavelet bases [1]. Namely, wavelet bases with rational dilation factor that share the same properties as Lemari´e-Meyer wavelet bases. Auscher has shown in [1] that the following result holds. Lemma 4.1 (Auscher).Consider an arbitrary rational p/q > 1, where p and qbeing relatively prime integers. Then there are functions ψ1, . . . , ψp−qin S(R)whose Fourier transforms are compactly supported and vanish in a neighbourhood of the origin, such that {(p/q)j/2ψl((p/q)jx−kq);j, k ∈Z,1≤l≤p−q}is an orthonormal basis of L2(R). We now introduce a wavelet decomposition with rational dilation factor for the random field {BH(t)}(t,H)∈R×[a,b]and for the MPRE {Z(t)}t∈R={BS(t)(t)}t∈R. Multifractional Processes 481 Proposition 4.1. Consider an arbitrary rational p/q > 1, where pand q being relatively prime. Then the field {BH(t)}(t,H)∈R×[a,b]can be expressed as the random series (4.2) BH(t, ω) = p−q X l=1 ∞ X j=−∞ ∞ X k=−∞ (p/q)−jH ×Ψl((p/q)jt−kq, H)−Ψl(−kq, H)l,j,k(ω), where {l,j,k}is a sequence of N(0,1) Gaussian random variables and where for every 1≤l≤p−qand (x, H)∈R×[a, b] (4.3) Ψl(x, H) = ZR eixξ b ψl(ξ) iξ|ξ|H−1/2dξ, ψ1, . . . , ψp−qbeing Auscher mother wavelets that generate a basis of dilation factor p/q. Then Ψ1, . . . , Ψp−qare C∞functions over R×[a, b] and their partial derivatives of any order are localized in xuniformly in H. Thus the series (4.2) is with probability 1, uniformly convergent in (t, H)on every compact of R×[a, b]. Proof of Proposition 4.1: This proposition can be proved by using the same techniques as in Section 2. Remark 4.1.The MPRE {Z(t)}t∈Rwith parameter {S(t)}t∈Rcan be expressed as the random series (4.4) Z(t, ω)= p−q X l=1 ∞ X j=−∞ ∞ X k=−∞ (p/q)−jS(t,ω) ×Ψl((p/q)jt−kq, S(t, ω))−Ψl(−kq, S(t, ω))l,j,k(ω), which is, with probability 1, uniformly convergent in ton every compact of R. Proof of Remark 4.1: This remark is a straightforward consequence of Definition 1.1 and Proposition 4.1. Now we are able to prove Theorem 4.1. Proof of Theorem 4.1: First we will suppose that ais a rational number greater than 1. We therefore have a=p/q,p > q > 0 being relatively 482 A. Ayache, M. S. Taqqu prime integers. As the stochastic process {S(t)}t∈Ris stationary and independent of the white noise it follows that for every x∈R, (4.5) (S(x), Z(x)) (a.s) = (S(x), BS(x)(x)) (d1) = (S(0), BS(0)(x)). Then Relations (4.2), (4.4) and (4.5) imply that for every t∈R, Zp qt(d1) =BS(0) p qt (a.s) = p−q X l=1 X j,k∈Z (p/q)−jS(0) ×Ψl((p/q)j+1t−kq, S(0)) −Ψl(−kq, S(0))l,j,k (a.s) = (p/q)S(0)BS(0)(t)(d1) = (p/q)S(t)BS(t)(t)(a.s) = (p/q)S(t)Z(t). Now suppose that ais a positive rational number lower than 1. We therefore have a=q/p,p > q > 0 being relatively prime integers. It follows from Relations (4.2), (4.4) and (4.5) that for every t∈R, Zq pt(d1) =BS(0) q pt (a.s) = p−q X l=1 X j,k∈Z (p/q)−jS(0) ×Ψl((p/q)j−1t−kq, S(0)) −Ψl(−kq, S(0))l,j,k (a.s) = (q/p)S(0)BS(0)(t)(d1) = (q/p)S(t)BS(t)(t)(a.s) = (q/p)S(t)Z(t). Finally suppose that ais a positive irrational number. Let (an) be a sequence of positive rationals converging to a. One has almost surely for every real t, aS(t)Z(t) = lim n→∞ aS(t) nZ(t)(4.6) and Z(at) = lim n→∞ Z(ant),(4.7) Multifractional Processes 483 because of the continuity of the process {Z(t)}t∈[0,1]. Since our previous results imply that for every n, (4.8) Z(ant)(d1) =aS(t) nZ(t), it follows from (4.6), (4.7) and (4.8) that Z(at)(d1) =aS(t)Z(t). We now give a sufficient condition for the MPRE to have stationary increments. Theorem 4.2. Let {Z(t)}t∈[0,1] be an MPRE whose parameter Sis a random variable independent of the white noise. Then the increments of {Z(t)}t∈[0,1] are stationary. Namely, for any t∈(0,1), one has (4.9) {Z(t+h)−Z(t)}h∈[0,1−t] (d) ={Z(h)−Z(0)}h∈[0,1−t], where (d) =means equality of the finite-dimensional distributions. Proof of Theorem 4.2: Suppose first that the random variable Stakes a finite number of values α1, . . . , αn. Since for any i= 1, . . . , n, the process {Bαi(t)}t∈[0,1] has stationary increments and is independent of S, it follows from Definition 1.1 that for any integer K≥1, reals θ1, . . . , θK, t∈[0,1], t+hk∈[0,1], k= 1, . . . , K, and for any Borel set D, P"K X k=1 (θk(Z(t+hk)−Z(t)) ∈D)∩(S=αi)# =P"K X k=1 (θk(Bαi(t+hk)−Bαi(t)) ∈D)∩(S=αi)# =P"K X k=1 θk(Bαi(t+hk)−Bαi(t)) ∈D#P(S=αi) =P"K X k=1 θk(Bαi(hk)−Bαi(0)) ∈D#P(S=αi) =P"K X k=1 (θk(Bαi(hk)−Bαi(0)) ∈D)∩(S=αi)# =P"K X k=1 (θk(Z(hk)−Z(0)) ∈D)∩(S=αi)#.